March, 1561
CONFIGURATIOX OF AN ADSORBED FLEXIBLE CHAIXPOLYMER
the experimental data. I n effect, the equations are insensitive EO values of v except a t extremely low concentrations where it is ordinarily not possible to obtain accurate data. The recent data on the adsorption of polystyrene on carbon of Frisch, et U E . , ~ illustrate ~ this difficulty clearly. The latter authors found that their data fit the Frisch and Simha isotherm better when an a priori assumed value of v = 50 was used rather than v = 1 (for a polystyrene of molecular weight 301,000). However, this yields a value of 900 A.2 area per adsorption site, which appears an order of magnitude too large in view of the probable area of not over 60 A 2 per segment as estimated from the data of Van der Vaarden.23 It still seems surprising that the adsorption data should reasonably fit the Langmuir isotherm in our case where v is so very large. For the (22) H L. Frisoh, M. Y. Hellman and J. L. Lundberg, J . Polgmer Sei., S8, 441 (1959). (23) M. J. Van der Vaarden, J . Coll. Sci., 6, 443 (1951).
487
polymers PLMA-3 and PLM-4-7, v g 470 and 1660, respectively. One might question whether the apparent fit to the Langmuir equation is entirely accidental. Simha, Frisch and Eirich4 point out that their isotherm is identical in form with the result for the adsorption of a v-mer with complete dissociation into “monomer” on the surface. This would appear to suggest that even in the case of the adsorption-desorption equilibrium of a polymer molecule, the equilibrium is basically a competition only between the individual adsorbing segments. We have studies in progress which may elucidate further the fundamental nature of the adsorption isotherm for polyfunctional molecules. Acknowledgments.-The authors wish to acknowledge the cooperation of Dr. D. G. Rea and hlr. H. M. White of our Spectroscopic Laboratory and are grateful to Drs. 0. L. Harle and W.I. Higuchi for helpful discussion.
EFFECTS OF SHORT RAXGE SURFACE-SEGMENT FORCES O S THE COXFICrURATION OF A S ADSORBED FLEXIBLE CHAIN POLY-VER BY W. I. HIGUCHI’ California Research, Corporatwn, Richmond, California, and The School of Pharmacy, Unicersity of Wisconsin, Madison, Wisconszn Recetved September 82, 1960
Following the model for the statistics of deposition of a flexible polymer chain from solution developed by Simha, Frisch and Eirich,2 equations were formulated which describe the segment density distribution as a function of a short range interaction parameter K’. Calculations for P(O),the fraction of anchor segments, were carried out over a wide range of K and the chain size. The calculations show that if anchoring of a segment is accompanied by a net change in potential energy, E, equivalent to that of about 4kT or more resulting from interactions of unspecific types, P ( 0 )may be essentially unity, i.e., the entire chain may be found collapsed to the surface. In the range 0 5 E 5 4kT, the changes in P(0)may be very large, corresponding to the change from a chain predominantly existing in the solution I‘ hase” to that which is collapsed to the surface. The recent experimental determination of P ( 0 ) by Fontana and Thomas for the system-poly(alky1 methacrylate) adsorption onto si1ic.a-is found to be consistent with the present treatment of the problem, for the case of large E .
Introduction with the expectation of finding a satisfactory The theoretical aspect of the problem of the quantitative description of the system within the adsorption of flexible non-ionic high polymers onto limitations of the model. The configuration of the flexible chain a t the solids from their solutions has been examined and discussed in rather great detail by Simha, Frisch interface is a very important facet of the over-all and Eirich12 Frisch and Simha,3 and F r i ~ c h . ~problem, as well as being interesting in itself. The model employed by these investigators in The present communication is concerned with the their quantitative discussions assumes that the effects of short range forces between the polymer adsorbed polymer chains are characterized by a segment and the active surface sites on the polyGaussian distribution of end-to-end distances. mer configuration. Frisch4 has already treated Knowledge of the average configuration of the the two limiting cases: very weak attractive polymer containing a statistically significant num- forces (energies > k T ) . It appeared to us statistical thermodynamics to the derivation of that the region in between these extremes should the adsorption isotherm equation. However, even be important for many real systems. for the simplified model, because of the complexity Theory of the situation, it is extremely difficult to enter Assume that the polymer chain in solution is the various interactions into a suitable equation characterized5 by its t segments of length A and a (1) The School of Fharmacy, University of Wisconsin, Xadison. Gaussian distribution of end-to-end distances. Wisconsin. Then with the adsorbing surface a t Z = 0 in the (2) R. Simhn, H. L. Frisch and F. R. Eirich, J .
[email protected]., 67, 584 (1953). x-y plane and anchoring the first segment of the (3) (a) H. L. Frisoh and R. Simha, zbid., 68, 507 (1954); (b) H. chain to the surface, the chain segment density, L. Frisch and R. Simha, J . Chem. Phys., 27, 702 (1957). (4)
H. L. Frisoh, J . P h w Chem., 59, 633 (1955).
(5) Conventions of Frisch, ref. 4, are used whenever possible.
488
W. I. HIGUCHI
Vol. 65
tive t o the solvent-segment the closer e would be to the actual energy of anchoring (ie., the minimum in the real potential energy curve). Because of the mathematical approximations, equation 2 is strictly valid for the model when s=l E > LT and foundthat in thelimit e+ >> kT, P(0)+1, P ( 2 )+O for 2 > R, Le., all t segments collapse to the surface layer and serve as anchors. An alternative approach but which essentially retains Frisch’s physical model will now be presented. This will permit using standard procedures for numerically computing P(0) for any E and large t. For the calculations it will be assumed as before4 that excluded volume effects may be neglected. This point will, however, be examined later in the discussion of the results. Also only short range interactions (range the order of one segment length) will be considered. Let us begin (see Fig. 1) with the first segment t Fig. 1.-A typical beginning of an adsorbed polymer (s = 1) of the polymer chain in Z = 0. The second chain of I, segments in the modified random walk representa- segment (s = 2) has the choice of residing in Z = tion (see text). 0 or in Z = 1 (as it does). Then the third seg(s = 3) has the choice to reside in 2 = 2 P ( Z ) ,may bt: calculated for the adsorbed polymer. ment 2 = 0 (as it does). This continues to s = t. or Here P ( 2 ) is defined as the fraction of t located between 2 and 2 -I- dZ. The coordinate 2 is The process is called random walk6 with a perfectly refiecting barrier if the probabilities for A 2 = in unite of A . This treatment for large t +1 is unity and A 2 = 0 is zero at 2 = 0. If, on the other hand, the probabilities for AZ = +1 and AZ = 0 are equal, the corresponding expreswhich is the expression for the fraction of t exist- sion for P(0) is, as will be seen later, equation ing as anchors for the polymer chain. I n the l w i t h a : = 1 a n d 6 = 0 . The effect of short range forces may be acequation a. is an accommodation coefficient (the probability of a successful contact between an counted for by suitable choice of the probabilities active site on the surface and a segment), .9 is the for a A 2 transition a t 2 = 0 and 2 = 1, maintainfraction of sites covered, and f is the segment ing equal probabilities of A 2 = 1 and A 2 = - 1 “diffusion” coefficient4 which includes the effects for all other 2 values. The walk process may be described by the folof the restrictions of valence angles and bond lowing differential equation under the usual rotation on the molecular configuration. To the extent that the discussion in this article limiting conditionss will be quantitative, it will be limited to examining (3) P(O), although a general expression for P ( 2 ) will he presented and discussed. Equation 1 is expected to apply to the case where where w = w(Z,t) is t,he end-to-end distribution no net interaction exists between the segment and function integrated over all .2: and y.’ The bethe surface sites. It is a consequence only of the havior a t 2 = 0 and Z = l follows the conditions geometric restriction placed upon the polymer chain. For the rase of very weak attractive forces Frisch4 obtained
P=
Z=
+
where X = e / k T , E is the net attractive surface- It is visualized, as indicated by these equations, segment energy, and IC and T have their usual for simplicity’s sake that there are three regions, meanings. More exactly, E is the depth of a 2 = 0 , Z = 1and Z 5 2. square well potential located a t 0 5 Z 5 R where The present approach differs from Frisch’s4 in R is the order of A for short range forces. Itj that the attractive forces are accounted for by the represents an average or “smoothed” potential use of conditions 4, 5 and 6 rather than by their energy decrease for the segment in 0 5 2 5 R rela(6) S. Chsndrssekhar. Reus. Modern Phys., 16, 1 (1943). tive t o the segment located a t Z > R. The less (7) This differs from the w ( 2 , t ) of the previous workers, ref. 1-4, by specific the interaction of the surface-segment rcla- the integration.
March, 1961
489
COXFIGURATION O F AN AkDSORBSDFLKXIBLT: C H k T S POLYMER
direct incorportition into equation 3 with Z 5 0. The other boundary conditions remain the same,*
viz. aut w, -&
--+ 0 as z--+
a,t
>0 (7)
and w
-+
6(Z) as t
--+-0
A few remarks on the meaning of conditions 4, 5 and 6 are necessary. The first term on the
+
right side of 4 is the “rate” a t which the (1 l)at segments are going to Z = 0 from the tth segments in 2 = 1. The second term in 4 is the “rate” l)stsegments are going to 2 = at which 1 he ( t 1 from the t f h segments in Z = 0. In condition 5 the first two terms on the right side are the l)8tsegments are coming “rates” ax which (t into Z = 1 from Z = 0 and Z = 2, respectively, and the third term represents the “rate” of segments having 2: = 1. Condition 6 simply represents mass balancing. The quantity K accounts for the effect of forces on these “rates,” i.e., the -04 0 04 00 I2 IS larger K is the I c w the tendency for the A 2 (0 to 1) and the A 2 (1 to 2 ) steps while the greater the LOG,, K , tendency for A Z {O to 0) and the AZ (1 to 2) Fig. 2.-A plot of P(O),the fraction of the total number pteps. A detailed examination of K will be given of segments t that are anchors, againRt the site-surface in the Discimion section. interaction parameter K . A, B and C correspond to 1 = To find a solution to the problem it becomes 1 X lo2,1 x lo3and 1 X lo4,respectively. necessary to make an approximation. From NOWP(0)is related to tut, 0 by2-4,9 condition (4) it i!; clear that,
+
+
P(0) =
From this it follows that for large t W-0
KWz-1
(9)
Tt can hc: shown that the effect, of not neglecting (dzu ’bt),=” i n equation 8, in the first approximation, is to effectively decrcasef by the order of ( K 2 3K 1 ) / ( K 2.f 2K 1) which is always sufficiently rlose to unity for our purposes. Employing w p t i o n 9, equation 3 may be solved8to gh-e
+
+
+
c
for Z 5 2
Thus u’i=2 =
2
(K
+ 1)2(b.-
a)
[chp(a*ft)erfc(a(ft)’h)-
exp(b2fb) csrfc( b ( f t ) ’ h ) ] (11)
S o w (dw ;dZ), :! may be obtained from equation 10 by differentitition and combined with 4, 5 , ti, I) and 11 to give E
U.,t=O
1
- a ) [(l - a ) exp(a9ft) erfc ( a ( f t ) ’ / 2 ) - (1 - b ) exp(blft) erfc{b(ft)’/zJ]
= -----
(1
+ L/Rj(b
(12)
( 8 ) H. S. Cardaw and J. C. Jaeger, “Conduction of Heat in Solids,” Oxford University Press, I.ondon, 1959, p. 306.
Jlt wI=o dl
(13)
It can be easily shown that equation 13 reduces to equation 1 for K = 1, provided a0 = 1 and 0 = 0 in equation 1. Also when K = 0. P ( 0 ) = 0. Results of Calculations and Discussion In Fig. 2 are given the results of P(0) calculations for three t values of 1 X lo2, 1 X lo3 and 1 X lo4 which should be representative of most real systems of interest (MW range of 1 X lo4 to 1 X lofior so). The calculations were carried out by a graphical method with equations 12 and 13, and an .f value of l/fi. This f value would correspond to a free chain (no valence angle rcstrictions and no b o d rotation barriers). In passing, it is worthwhile to mention an alternative interpretation of the meaning of the results of Fig. 2 with f = l / 6 . If the concept of the statistical chain element of W. Ihhn‘O is introduced to characterize the polymer chain, then f = 1/6 mill include the restriction of valence angles and bond rotation. I n this interpretation the number, t’, of the statistical chain elements (of length A’) would be less than t for a given polymer chain according to the following conditionslO
1 i
t’ = t / n A‘ = r.m.s. length of n segments of length A where n is large enough to make orientation of one element independent of the (I4) next At = A’t‘
It is clear that in some respects the statistical chain element analysis would be more consistent (9, I n eq. 12 70-0 is the same as p(r\ used by the p,evious aorkers. See ref. 7. (10) See P. J. Flory, “Principles of Polymer Chemistry.” Cornell Vnirersity Press. Ithaca, N. Y., 1953.
W. I. HIGUCHI
490
with the model of random flight. However, in the present discussion, A and t will be used as b e f ~ r e ~t-o- ~describe the polymer chain for the reason that a clearer relationship between the variables can be seen with this choice. The long range effect on P(0) of increasing f while holding K constant would be essentially the same as proportionally increasing t (see equation 12). For linear polymers involving the tetrahedral carbon linkages f is the order of 2 to 4 times greater than f for the free chain. This would correspond to about a 10 to 20% downward shift in the results of Fig. 2. For more inflexible polymers the increase in f would be greater. The effect of solvent on P(0) while holding K constant is to egectively either increase or decrease .f depending on whether the solvent is a good one or a poor one, and on whether there is n high concentration of other polymer chains nearby (intermolecular excluded volume effects). The discussion up until now has assumed the Flory theta solvent, ie., the net effect of solvent-segment and segment-segment interactions was to permit the random walk calculation of the configuration to be the correct one with f providing the valence mgle and bond rotation restrictions only. 'The Flory type treatmentlo involving the expansion factor, alpha, may be used to estimate" the effective change in f caused by solvents. For linear polymers the effects of solvents are about the same order as in the three-dimensional cases,l0 as perhaps expected. For not too extreme solvents the effective changes in f over the f for theta solvent is generally the order of 2 or so, even for moderate coverages. At very high coverages, when both inter- and intramolecular excluded volume effects could become very serious, the effective f will be much larger. It appears that K , itself, might be the most important I'(0) determining factor, especially at low to moderate coverages. Particularly for the high molecular weight polymer chains, K has a Tery lmge influence on P(0). For example (Fig. 21, taking the t = 1 X lo4 chain, while at K = 1 P(0) is -0.03; at K = 3, P(0) = 0.13a fourfold eflect; and at K = 10, P(0) = 0.6a twenty-fold effect. At K 7 30, polymer chains are essentially collapsed to the surface. The character of an adsorbed film undergoes drastic change:; in the region of 1 K 30. The segment density distribution, P ( Z ) , which could be calculated along the same lines as the P(0) calculation by means of equations 11 and 13, should reflect these changes. The adsorbed film thickness, ho\Yever defined, should be considerably smaller for K = 30 than for K = 1. The meaning of K now requires discussion. From equations 4 and 5 it is seen that K is a kind of distribution coefficient for Z = 0 and Z = 1. When i,he ttkl segment is in Z = 0 the quantity K / K $- 1 represents the probability of anchoring for the [ t l ) s segment; t while 1/K 1represents the probability for not anchoring. Now we have also chosen K / K 1 to represent the probability
- -
+
+
+
(11) Unpub1ish:d results employing a onedimensional modification of I'lory's mrthod
Vol. 65
+
of anchoring of the (t 1)st segment when the t t h segment is in 2 = 1 with again letting 1/K 1 represent the probability for not anchoring. Strictly the same probabilities should not be used for the two processes, or, alternatively, different K's should be used. The problem arises because even though the potential energy change for anchoring may be the same in the two processes, the degree of orientational freedom associated with anchoring differs (neglecting restrictions of valence angle and bond rotation). However, the effect is small enough to be unimportant1? for the present discussion and thus the simple derivation will be retained. We may write K
ffoKo(l- e)
=
and
+
1
(15)
KO = exp (&T) and (YO, 0 and E have already been defined. For e/kT
