A. SILBERBERG
1872
A. H. ELLrsoN.-The 11-A isotherm is reversible up to a pressure of II,. Above the pressure of IT, the film i s unstable and shows hgsteresis on dccornpression If the film is expanded so that t,he pressure falls to or below ITe and then recompressed it aould be as though a new film had been spread and as stntpd in the paper not lilcelj to yield a good reprodurfion of the isotherm above TIF nbiained in the first, rompression.
J N. WILSON (Shell Development Company).-The superiority of acid washing over distillation for the purification of mercury suggests that polyatoniic clusters exist in mercury vapor, especially a t relatively low temperatures. Because of the relatively low specificity of metallic bonding, such clusters may act as carriers t o transport metallic impurities through the vapor.
Vol. GG
F. M. FOWKES (Shell Development Company).-May I suggest that the pressure-area isotherms of Fig. 1-4 might be easily expressed in terms of the equations developed in paper I1 of my series on Ideal Two Dimensional Solutions 1.I. Phys. Chem., 66, 386 (l962)],treating the monoluyers ( a t least up t o 11 = 11,) as surfaces of mercury atoms (component 2) diluted by organic molecules (component 1'1 which should not be associated on this surface because t h w are more strongly bound to mercury than to each other [see J , Phys. Chent., 66, 383 (1862)l. The film pretrsure II ahould he a function of the mole fraction of mercurv atoms in the film-covered surface (Xl)and the partial atomic area of mercury ( a 2 ) ,
11=
-kT In X , uz
THE ADSORPTION OF FLEXIBLE ITACROMOLECULES. PART I. THE BY A. SILBERBERG~ Mellon Institute, Pittsburgh, Penna, Received March 0, 1068
The adsorption of an isolated, flexible linear polymer molecule of high molecular weight is treated a t an infinite plane surface. The question of a possible equilibrium configuration is examined when it is required that some a t least of the segments of the polymer molecule are in contact with the surface, and it is assumed that for each segment so placed the internal energy of the system is reduced by an adsorption energy zkT. It is shown that the polymer molecule will split up into sequences of segments; alternate stretches of PS segments all in the surface and loops of PB segments all out of the surface, whose size is not a function of molecular weight. The length, Pg, of the loops will decrease and the fraction, p of segments in the surface will increase as the absorption energy z increases. Several models are considered. For an all adsorbable polymer molecule on an all adsorbing surface, the loops P g are small and p ie large even at small values of x, Such polymer molecules stay close t o the surface with practically all their segments, and behave essentially as two-dimensional structures. In cases where not all surface sites are adsorbing, or not all polymer segments are adsorbable, or both, the size of loop is considerably increased and much higher adsorption ener8ies are required, Restrictions limiting the re-entry of an adsorption loop into the surface, as well as any increase in specificity, have a similar effect. It is shown that structural principles affect polymer adsorption in sensitive fashion. The methods described can easily be adapted t o discuss a variety of cases and are useful for this reason. It is one of the !onsequences of thia model that the enda of the polymer molecule are on the surface. A discussion of previous theories, which predicted rather different results, is given.
1. Introduction polymer and surface characterized and held conWhen a flexible polymer is placed near or onto a stant, the thermodynamic equilibrium between surplane surface, great distortions of its shape from the face phase and bulk solution was considered. Unaverage are to be expected which should depend in satisfactory formulations were used, however, for sensitive fashion on the structural features of the both these aspects in the original paper^,^^,^^ and, chain and on the chemistry and reactivity of its although the thermodynamic part has been corgroups. Adsorption is thus an int'eresting tech- rectede and the treatment of the isolated mo!ecule ~ , ~ remain two serious weaknesses nique to be employed in polymer st,udies,but an ade- i m p r o ~ e d ,there quate theoretical apparatus must' be developed to in the treatment: the discussion of the isolated macromolecule should be re-formulated and the make sense of the results obtained. Most of what has been done to date3-7 in this shape of the polymer must be introduced as a varidirection is based on a set of fundamental p a p e r ~ ~ ~able v ~into ~ the full thermodynamic treatment. In the present series of papers we shall attempt to by Frisch, Simha, and Eirich in which t'he problem was first' stated and analyzed. Theirs was essen- do both. In Part I we shall discuss the isolated tially a two-step approach, involving in the first macromolecule itt a plane surface. We shall coninstance a discussion of the shape of the polymer in Rider first cases where all segments of the polymer t,he presence of the surface. Then only, with the can adsorb and where all sites on the surface are shape and the degree of intimate cont,act between adsorbing. We shall consider these cases under a variety of structural restrictions on the chain as it (1) This work was supported b y the Air Force Office of Scientific enters or leaves the surface. In addition we shall Research under contract No. 4 F 49(638)541 with Mellon Institute. consider cases where not all surface sites are ad(2) On temporary leave of absence from Weinmann Institute of Science, Rehovot, Israel. sorbing in character and where not all polymer seg(3) (a) H. L. Frisch, R . Simha, and F.R. Eirich, J. Chmm. Phys., 21, ments can be adsorbed. Emphasis will be put on 36.5 (1953): (b) R. Simha. H. L. Frisch, and F. R. Eirich, .I. P h y s . the method of derivation and it will become clear Chem., 6 7 , 584 (1953); ( c ) H. L. Frisch and R. Simha, i b i d . , 58, 507 from the discussion that these methods can easily (1951). (4) €1. L. Frisch, ibid., 59, 633 (1955). be extended and applied to a large number of other (5) H. L. Frisch and R . Simha, J . Chem. Phys., 24, 652 (1986). cases not considered here explicitly. In Part TIs we II. L. Frisch a n d R. Simha, i h i d . , 2 7 , 702 (1957). (7) W. I. IIiguchi, J . r h u s . Chem., 65, 487 (1961). (6)
(8) A. Silhrrhrrg, abid., 66, 1884 (1962).
Oct,., 1962
ADSORPTIOK OF FLEXIBLE MACROMOLECULES
shall treat the over-all thermodynamic aspects and derive expressions for the adsorption isotherm, for the surface tension decrement, for surface pressure, and for the parameters characterizing the structure of the monolayer. 2. General Formulation of the Problem.In all the present considerations we shall regard the polymer molecule as being of high molecular weight. We place the polymer molecule in the vicinity of a plane surface and ask what Configurations it can adopt when a t least some of its segments are required to be in contact with the surface. It is clear that the polymer molecule will split up into a series of runs composed of segments which are either all in contact with the surface, or all out of contact with it. In general let there be m, runs of z segments each all out of the surface and hj runs of j sepmenta each all in the surfaae, The variables i and j can adopt any value, but if there are P segments all together in the polymer molecule, and we ignore eIid effects, the values which may be given to m, and &, are subject to the restrictions
1873
(3%) where the equilibrium values of m, PB, and Ps are fixed by the requirement that they determine the maximum term of Q. Kote that m, PB, and Ps are not independent of each other but must conform to the accessory conditions (1) and (2). Condition (2) has already been introduced into (3a), and (1) reduces to
+ Ps)
P (4) under present circumstances. It is appropriate to consider the logarithm of the term in Q and replace the factorials which occur by the Stirling approximation. We thus have rn(PB
In Qterm
= w(PB
mPB In
=
+ Ps) In 9 + m In w& +
PB Pa + mPs In Ps 3. - 1 - 1
P B
m In ( P B - ~ ) ( P s 1 ) (5) Multiplying ( 5 ) by a Lagrangian multiplier X we define a function
and
2mi = Zm., (a) = In &term Xm(Pn Jr Ps) (6) We then define functions, w ( i ) and W ( j ) , representing the thermodynamic probabilities of a and consider the maximum of Q^ with respect to m, run of i E>egments,in a loop near the surface, and a PB, and Ps, now regarded as completely independrun of j segments, totally in the surface, respectively. ent. This leads to a set of three equations from The functions w and (;1 are counts of all the distin- which X may be eliminated to give guishable configurations which can be adopted by a sequence of segments under the restrictions imposed, weighted with any Boltzmann energy factor which might arise. w ( i ) and G ( j ) are functions of i and j , respectively, but are independent of m(i) b In w + I n ; - P s - b In ij and m ( j ) . Their functional structure is fully de- In w - P B bPB bPS termined by the restrictions imposed and may be presumed known. What they look like in detail In (Pn - 1)(Ps 1) = 0 (8) will he discussed in later sections. If q represents the partition function over all as the set of equations to be solved for PB and Ps internal dpgrees of freedom of a polymer segment, simultaneously. To test whether the solution of (7) and (8) corvcie may write the following expression for the responds to a maximum we must look a t the second partition function Q of the entire system derivatives of We note that, of these, only r r t m l + rimJ Dw(i)mn6(j)KJ x Q 3 2 ~ / band P ~d2Q/bPs2 2 are non-zero. As necessary m,m conditions for a maximum we thus find
+
4
+
~
-
6.
Ciml Cjk, (=mi >(E%,
)
(3) It is our task to determine the equilibrium values of i, j , nz, and @z. What we expect to find is a sharp maximum in m and m a t a certain value of z andj, respectively, with the functions zero practically everywhere else. By way of assumption, therefore we shall write m ( i ) = m6(i - PB)
& ( j ) = m6(j - Ps) where 6 stands for the &function, PB pnd PS are the equilibrium values of i and j , respectively, and m is the total number of runs of each kind. With these assumptions Q reduces to
and
Note that we would still be dealing with an overall absolute maximum if in one, but not in both, of the equations 9 and 10 the inequality were replaced by an equal sign. If this is the case (and let us assume that PB reaches such a point) the nature of the solution is altered. PB must now be regarded as constant and in place of (7) and (8)we find a single equation to be solved for PS
A. BILEEREERG
1574
112 ( 2 - s i
a
Vol. 6G
X Lattice first and simplest case we shall consider the type of lattice illustrated in Fig. l(a) and (b). Each site may be occupied by a chain segment and is surrounded by Z nearest neighbors, Le., there are assumed to be only 2 possibilities of arranging a bond between t.vi.o adjoining chain segments jn the bulk of the system. On the surface, for a chain constrained to stay entirely within it, the coordination number reduces to S. In the vicinity of the surface we may regard the lattice as an array of layers of sites similar to the surface layer with (2 - S ) / 2 ways of stepping from one layer into the next and S possibilities of staying within it, With these definitions, the problem of counting the number of arrangements on this lattice for a random walk, which sets out from the surface and returns to it after a certain number of steps, without any of the intermediate steps reaching the surface, can be handled in principle. In particular, if the number of steps is small, there is no difficulty about writing down the number of arrangements explicitly. We recall only that an adsorption loop of Pg segments corresponds to a random walk of Pn 1 steps from the surface and back into it, and that we adopt the rule that in the placement of the step from a segment IC to segment ( k 1) only the position of segment (k 1) will exclude a site. In this way, we arrive a t the results of column 3, Table I, where cases up to PB= 5 have been treated. The manner of derivation should be obvious. Let us now consider the expression 3. Configuration Count on a 2 -
(Hexagonal Lattice).--As
lb)
la)
+
(C)
Fig. 1.-Schematic representation of lattice considered: (a, b) hexagonal lattice; (c) cubic lattice.
a In W In 6 - (PB* Pa) -
+
X
?IPS
Ps - 1 iIn ( P B * - 1 ) ( P g 1) PS In w(PB*) = 0 ; PB = PB*= const.
-
+
- + (11)
where P g * is the value of PB corresponding to an equality sign in (9). A somewhat similar situation would arise if the physical restrictions of the model require one or the other of the variables PB or Ps to be constant. For example, the structure of the surface could conceivably permit only single point attachment. Ti1 this case, Ps = 1 and is constant, and we find
Putting 2 I2 and X = 6 and equating (14) with the results of column 3, Table I, we can calculate f in each case. The values f found in this way are given in the last column in the table. It is seen therefore t,hat expression 14 with f = 0,7 would be a reasonable representation of w ( P B )over the range considered. As it mill turn out, moreover, that by the use of (14) we arrive at values of PB which fall within the tested range, we may regard (14) with f = 0.7 as adequate to our purpose. For W,as is easily seen, we may put
i;(P,) = S(S In W (1) = 0 ; Ps
=
1 (12)
as the equation for PB. Note that knowledge of PB and PS gives complete informat,ion about the state of the polymer. We shall in particular be discussing PB and the fraction p =
Ps P T Z
of polymer segments adhering directly t o the surface. We want to study these quantities as functions of the model introduced and the parameter values chosen. Explicit forms for w and G must thus be established for each case.
1)Ps - 2 e x P s
(15)
where x is the segment adsorption energy, i.e., the reduction in internal energy (in units of IcT) per segment occupying a surface layer site. For PB and Ps we thus derive the following equations from (7) and (8), respectively Ps P B In -~ = In ___ Ps - 1 PB - 1
In
1
__(PP
In
- 1)
i(f
=
(Z
-1 + In 2-__ + lnf 8-1
In ( P B - 1)
-
-
2
+
S ) ) ' S / ( Z - 1)(X - 1)2] (17)
NUMBER OF No. of segments
in loop,
PB
TABLE I ADSORPTION LOOP ON b
COBFIGURATIONS OF AN
Max. no. of layers involved
- 8 LAWICE I
Total no. of configurations w(PB)
0.67
1
3
re)
.66
2
2
(? s 2? -5 ) 2 -( 8 ~ I)('+) ) ( +y +(')
S(S
-
1)
-72 .72
4
5
I
We note moreover that the conditions (9) and (10) will be satisfied provided PB and Pg are larger than 1. The results for the numerical case 2 = 12, S = 6, are shown in Fig. 2. We note in particular that PB IS smaller than 5.5 €or positive z and that more than half the segments will be in ihe surface ( p > 0.5) when the segment adsorption energy z exceeds l / ? ( k T ) . The polymer chains are thus flattened into the surface a t relatively small segment adsorption energies, and the long loops predicted by the Frisch, Simha, and Eirich model do not occur in this case. Furthermore, the dimensions of the loops are functions only of the lattice parameters 2, S, a n d f a n d the adsorption energy 5 . The total length of the chain P does not influence the character of the solution, provided that it is large enough. 4. Configuration Count on a Cubic Lattice.-(a) The lattice which we have considered in the previous section is the one customarily used in polymer solution theory. For the consideration of a variety of restricted random walks, such as we now propose to do, it is not the most suitable one, however. For this purpose we do better if we consider the cubic lattice illustrated in Fig. 1(c). For each step on this lattice there are two choices oplen in each of the principal directions, i.e.. with each step we progress either to the left or to the right, up or down, backwards or forwards, by a constant amount. Which choice is taken in each direction ic;independent of the decisions taken in the other two directions, or, in other words, the random walks along the principal axes of the lattice are independent of each other, Let us therefore divide up our problem and consider the random walks, along the normal to the surface layer, and parallel to it, separately. We thus look for the number of configurations (cr, and wp normal rind parallel to the surface, respectively, in the knovvledge that
~ ( P B=) w ~ ( P B ) w ~ ( P B ) (18) for any combination wn and up referring to the same numher of isteps. A similar equation holds for G.
0
0
05
IO
I I 15
20
25
70
0
Adsorption Energy x (Unlts of kT).
Fig. 2.--Length of adsorption loop P B and fraction of number of segments p in the surface at various values of the adsorption energy 2 on a 2 - 8 lattice; 2 = 12; S = 6; f = 0.7.
(b) We have shown the progress of a linear random walk in a direction normal to the surface in Fig. 3(4. Along the abscissa we have plotted the position relative to the surface, in units of lattice distances. (One lattice distance corresponds to half the length of side of the cube shown in Fig. 1(c).) Along the ordinate we plot the ordinal number of the segment under consideration. The one dimensional random walk is thus pulled out, concertina fashion, along the ordinate of the figure for our more convenient inspection. The lattice extends from m on the abscissa to position 1 where the outermost layer of lattice sites is assumed to occur. Positions larger than 1 are empty of' lattice sites. The lattice sites at position 1,when occupied by a polymer segment, reduce the energy of the system by an amount xkT for each site filled. If a stretch of polymer is totally adsorbed the restrictions of the cubic lattice imply that only every second segment can be a t layer line 1. The in-between segments must return to layer line 0 and thus will not contribute an energy change. What we shall call a random walk in the surface is in fact a succession of alternations between posi-
-
A. ~ I L B E R B E R G
18’16
tions 0 and 1 in Fig. 3(a). A totally adsorbed stretch has only one configuration along the normal to the surface and if it in~~olves Ps segments will conlxibule an internal energy change of (x/2)(Ps 1 ) units of IrT. (Sotc., that, by definition, an adsorbed strctrh starts and elids in layer linP 1,) For &(Ps)wc thus have
+
In the case of an adsorption loop of PB segments we know that it starts in layer line 1 and returns to it in Pg 1 steps. While we must exclude all occupations of layer line 1 by these PB segments, there is no objection to layer line 0 being occupied, As shown in Fig. 3 the positions of the first step AB and of the last step CD are fixed by these considerations. The (PB 1) steps between B and C may be taken anyhow, but are not allowed to reach layer line 1. One of many permissible paths is shown as the dotted line in Fig. 3(a). To count their number we consider all paths betFTeen B and C but place an adsorbing ~vallat layer line 1 to remove, from the total, all paths which touch it. According to Chandre~ekhar~ we can find this number if from all paths between B and C we subtract the paths from B to the mirror image point C’ of C, taken with respect to the wall, We thus easily conviiice ourselves that
+
-
Trol. 6G
sites. In such cases it is not necessary to exclude the outer layer to paths between adsorption points as no energy changes are involved. The situation corresponding to this model is illustrated in Fig. 3(b). The physical reality of this model depends on the assumption that, given a loop of any length between two adsorption sites, no other adsorption point can be reached by the paths between them. We ahall later examine how valid this assumption can be made, when this model will be discussed in detail. r o w we are concerned only with establishing u,(PE) for this case. Note that W,(Ps> for thir, model is simply Ljn(Ps) = ij(P,) = ex (22) To determine w,(Pn) w e proceed &B before, for the first and last step, but place the adsorbing wall a t position 2 (and the mirror image point C’, consequently, at 4)when considering the configuration of the intervening (Pp, - 1) steps. We find
2pB
+3
(2T)’”(PB f 3 ) ( P B f 1)’”
(23)
(e) In assessing the influence of the surface in restricting configurations it should be recalled that the total number of configurations for PB 1 steps in one dimension is 2 h + I . The restrictions which apply to equation 20 thus reduce this number by a factor ( 2 T ) * ” ~ E ( ~ E I)~’$. In terms of entropy this represents a decrease by an amount k In [(27r)’)gPB(PB 1)’/2] for each loop of length P g . Note that this is in fact relatively little. Even if Pg is large, say of order 100, this loss in entropy could be compensated for by a gain, in other ways, of about 8kT of energy. If, moreover, we allow for the additional freedom implied by equation 23, the entropy loss in a loop of about 100 segments would be counterbalanced by an even smaller energy change of about 6.6kT. It should be clear therefore why long loops do not, as a rule, survive the adsorption process if a small energy gain per segment may be made in the surface. (f) To round off this discussion we must find out what happens in the directions parallel to the surface and establish up and &. As the simplest assumption to make we shall consider first that the random walks in these directions are completely uninterfered with. Under these circumstances
+
+
+
2 p B -k 1 (27$’2PB(PB
+ 1)l’Z
(20)
where use mas made of the Stirling approximation to represent the factorials. (c) Using the same approach it is also easily shown that the number of paths which go from the surface, but, in 1 steps, do not necessarily return to it, ;.e., may reach any point from the surface without touching 011 passing through layer h e 1 (except perhaps with the last step), is given by
q,(Pn)
2 2 ( p B f 1)
&(PS)= 2 W S - 1) 21 -. 1 (2TC)”2’
I
>> 1
(21)
(d) We shall in addition be considering cases where not all outer layer line sites are adsorption (9) 8. Chandresekhar, Rev. Modem Phys., 16, 1 (1943).
(24) (2Ci)
(g) To assume such a measure of freedom may, however, be rather nayve. It should be realized that we have put no restriction on the configurations, resulting either from multiple occupancy of the same lattice site or from steric interference. As a small step in this direction, and in order mainly to test out the effect of having some such restric-
ADSORPTION OF FLEXIBLB MACROMOLECULES
Oct., 1962
1871
tiona put, in, we shall consider limiting the possibilities of return into the surface. What we have in mind is illustrated in Fig. 4 ( 4 . The adsorption loop depicted there is restricted by the assumption that the outer layer surface sites within the shaded area cannot be occupied by the returning loop. The ends of an adsorption loop are thus constrained to be a t least d lattice distances apart. We have already made use of the fact, that the number of configurations for %I random walk of (Ps 111steps, consistent with an end separation of 6 lattice distances, is given by
+
(Pa I__-
pB
+ ;)!
+
(-5-
+ I)!
(-- + 1 - !)I P B
2
L
2
-4
angles to it, such that
fZ2
=
A
+
'
;
$/l
, 2
4
(0)
If there ig a separation +t1 into one direction and .ttl into another, independenf direction, at right
+
-2
n
-2
P
the number of configurations consistent with a separation thus turns out to be
2
0
Position Normal to Surface in Units of Lattice
/
612
b
,/i
or, using the Stirling approximation, by
\'.
l
I
I
4
6
DisIances. (bl
Fig. 3.-Schematic representation of a linear random walk normal to the surface. The random walk is pulled out, concertina fashion for inspection, nFith each step findings its appropriate locakon along the ordinate. The lattice &-etches from - m to layer line 1which is thus the physical surface. The steps AB and CD must be taken as ahown. The ( P B - 1) intervening steps may be taken in any way which does not take them up to or beyond an absorbing wall (shoxrn shaded) and placed as indicated. A possible permitted path is shown dotted. (a) Occupation of surface layer sites is forbidden; (b) occupation of surface layer sites is permitted.
by (26) and since there are 8n points a t distance nD we have
L,
where the factor allo.crrs for the fact that there art: four mays of arriving at I$]. Expression 26 is thus 1,he density of configurations reaohing a point a distance 4 from the origin of the configurations. If we integrate (26) over the whole area we find, ax expected, that
(28)
It is obvious that the above is a very rapidly conrergiiig series, proyided
(PB-k 1) < 0' (29) We shall break it off therefore aft>erthe second term, and thus find
Le., the integration gives the t o t 4 number of configurations (24). Here we wish to assess the number of configurations with end-points falling outside a distance d from the origin. Applying (26), we thus find, after suitable integration, that
up(PB)= 2 2 ( p B + 1) e - - d 2 / 2 ( P B
+ 1)
(27)
(h) Another powerful restriction of the random walk parallel to the surface would oecur if not all outer layer sites were adsorbing. We could visualize, for example, that only layer points D lattice spaces from each other couId adsorb with an energy effect (Fig. 4(b)). As can be seen, points A and B of the adsorption loop shown in the figure are in the surface, but they are so without an energy effect. The end-point of a loop of PB segments will thus be nD lattice distances from the origin where n = 1,2,3, . ., (PB l)/D. The number of configurations lea,ding to a point for which = nD are given
+
'ITie can now proceed to a discussion of explicit cases. 5. Cubic Lattice; All Polymer Segments Adsorbable ; All Surface Sites Adsorbing ; Return into Surface Restricted.--As all surface sites are adsorbing we cannot permit interior segments of adsorption loops to enter the outer lattice layer. Hence we shall be using (19) and (20) for W, and war respectively. For Ljp and up,in keeping with the restrictions envisaged, we shall put (25) and ( 2 7 ) , respectively. We thus have
and
for subst,itution into (7) and (8). These are then solved for PB and Ps. Examining the restrictions (9) and (10) we find that the solution is valid for (PB
Big. 4.-Restrictions on adsorption loops in a cubic lattice. The diagrams visualiae the outer layer line surface Fites (corresponding to position 1 in Fig. 3) seen from below, The lattice site positions occur at the intersections of the lines as shown. Each such square corresponds to t8heside of the cube in Fig. l ( c ) and its edge measures out 2 lattice distances. (a) The ends of the adsorption loop are iecluded from being closer than d lattice distances apart. ?he area shaded is excluded to the returning path. (b) Not 911 aurface layer sites are adsorbing Those which are, are shown as open circles and are D lattice distances apart. Points A and B may occupy surface sites without energy change. (c) Not all polymer segments are adsorbable. Those which are, are shown as open circles in the chain and are A‘ segments apart Point A is in the surface without energy change. Point B, however, represents an adsorption. MAXIMUM LENGTH OF ADSORPTION LOOPS. 100.
1
Polymer N = I O
80-
The Occupation of Surface Sites By Internal Segments of Loop
40
+ 1) [1 -
]
- 1) < 2d2
2 ( p B+ PB2(PB
(33)
only, so that, depending on the value of d, the number of segments in an adsorption loop may not exceed a certain value PB*. How this number depends on the parameter d is shown in Fig. 5, curve 2. Curve 1 refers to a hypothetical case, analogous t o the above, where instead of (20) for Wn we have used (23). Internal segments of the adsorption loops may enter outer layer sites in this case. As such contacts would be energy contributing, this is not a realistic model and was plotted only for the interest the curve possesses when the definition of the parameters is changed. We shall discuss this presently. The parameters p and PB are plotted as functions of z in Fig. 6 and 7 as curves 2 and 3 for the cases d = 2 and d = 4,respectively. In Fig. 6, in addition, we have considered the case d = 8, as curve 4. Points a t energy values 2 above the break refer to PBvalues below the critical given by (33). Points below the break are a continuation of the solution by the use of equation 11. They represent a gradual dissolution of the adsorbed stretches into loops of constant length PB*. It should further be noted that we must have Pg > d 1 (see Big. 4(a)) as otherwise the adsorption loops could not span the excluded area, The curves for d = 4 and d 8 in Fig. 6 and 7 thus come to abrupt stops beyond which adsorption is complete on this model. Note moreover that the abscissas in Fig. 6 and 7 have different meanings, although they refer to the same model. In Fig. 6 we have averaged x over the adsorbed segments, including those a t layer line 0 in Fig. 3. For the cubic lattice model, now discussed, Le., for curves 2, 3, and 4 in Fig. 6, the abscissa is thus z/2. In Fig. 7, on the other hand, we were interested in comparing the actual binding strength of the surface interactions. The abscissa is thus x in all cases. It is interesting to note that increases in d sharpen up the rapidity of the transition in the value of p . As curve 4, Fig. 6, shows, an energy change of 0.5kT brings about a change in p from 0.03 to 0.97 when d = 8. Note moreover that curves 1 and 2 in Fig. 6 which refer to entirely different lattices agree with each other approximately, except a t high p values where PB is small and the models may be expected to diverge. As pointed out in section 3, p is large, for posj tive adsorption energies, and reaches about 0.75, when the energy change per segment is one IcT. This is true for both curves 1 and 2 in Fig. 6, which refer to the least specialized conditions. We have so far assumed that temperature is constant and that the different values of x correspond to different systems, If A E A d s is the energy change in the adsorption of a segment, in conventional energy units, we have
-
I +
OO
1
2
3
4
.
Excludea Surface Parameter d Fig. 5.--Maximum length PB*,of adsorption loops for various values of the excluded surface parameter, d.
RD~ORPTION OF FLEXIBLE MACROXOLECULES
Oct., 1962
1879
FRACTION OF POLYMER SEGMENTS ON SURFACE.
(34) and we may regard x: as an inverse temperature, if AEAds is constant. The abscissas in Fig. 6 and 7 are thus proportional to 1 / T , and the changes observed may be interpreted as a gradual melting out of adsorbed stretches of segments and their expansion into loops, which increase in size with increase in temperature. The breaks thus mark transitions in the state of the polymer mo1ei:ule a t the surface. 6. Cubic Lattice. All Polymer Segments Adsorbable ; Not All Surface Sites Adsorbing ; Return into the Surface Unrestricted.-We are discussing the situation depicted in Fig. 4(b). No totally adsorbedl stretch larger than one segment can occur. We are thus dealing with the case envisaged in the last part of section 2 . Pe = :L and we have equation 12 to solve for PB. Remembering that under the conditions proposed \+-emust use (22) for W,and (23) and (30) for w, and up, respectively, we have
I
06
P.
’
04
02
b-:/
, l,’4
I Hexagonal Lottice z = 12 ~ = 6 2 Cubic Lattice 3 Cubic Lattice 4 Cubic Lattice d.8
I
I
I
LENGTH OF ADSORPTION LOOPS
(35)
&(l)= ex and
wit,h
Y
-=
[ --]
exp -
0 2
2(PlB
+ 1)
(37)
Substitution of these into (12) then gives the desired solution. The restrictions (9) and (lo), moreover, must be satisfied and it turns out that these requirements are stronger than (29) so that the assumptions made in deriving (30) will always be consistent with the above solution. The values which P B can assume are bounded. We must, have Pg > D - 1 (see Fig. 4(b)). On the other hand the restrictions (9) and (10) set on upper bound to P B , which may be read approximately from curve 1, Fig. 5 , if we interpret the abscissa in terms of D, according to
Por the case D = 10 the solution is plotted as curve 5 in Fig. 7. S o t e the big shift along the energy axis and the large P B yalues which are obtained in this case. If x = 6, for example, Pg increasesfrom about 1 (on a uniformly adsorbing surface, curves 1 or a),to about 40, in the case here considered. Apart from the approximations leading to equation 30 we have one further source of inaccuracy in our derivation. We have permitted internal segments of loops to occupy surface sites, as is logical in this case. This, however, does not preclude the possibility that an adsorption site, i.e., an active surface site, is on occasion occupied by an
Adsorption Energy x (Units of k T )
Fig. 7.-Average number of segments, PB, in an adsorption loop in dependence on the segment adsorption energy, X
internal loop segment. Such configurations (which we have included in the count) are not permitted. Their number, however, is only a fraction of the order 1/D2 of all the configurations and will be negligible if D >> 1, as is here assumed. 7, Cubic Lattice; Not All Polymer Segments Adsorbable ; All Surface Sites Adsorbing ; Return into the Surface Restricted.--We have so far assumed that the segments into which the polymer is divided are both of the right size to occupy one lattice site and of uniform adsorption properties. We shall now treat a case where the ability of the polymer to adjust to the lattice is unchanged, but only every Nth segment will, when in the surface, make an energy contribution. All other segments may occupy surface sites, but this occupation does not involve an energy change. Figure 4(c) illustrates what we have in mind. Only the segments indicated by open circles are adsorbing. The section of polymer chain we have shown consists of two loops, as only segment B, but not A, is adsorbable and changes the energy _.of the system. From thc point of view of the adsorption process, we shall, nom, find it conveniciit to define as an I
1880
A. SILBJCRBERG
adsorption segment only the adsorbable segment, ignoring for the moment its (LY- 1) immediate non-specific neighbors. In Fig. 4(c) we have shown 5 such adsorption segments, The first one in the surface, then a sequence of two, out of the surface, and then a sequence of two again, in the surface. The treatment of section 2 can thus be applied substituting only adsorption segments for what me have called segments before. We will indicate with primes all symbols, previously used, but which now refer to enumerations in uiiits of adsorption segments . We thus have the polymer adsorbed in stretches of P p ' adsorption segments, all in the surface, and in loops of PB' adsorption eegments, all out of the surf ace. ,411adsorption stretch is composed of (Ps' 1) loops of ( N - 1) segments each. For each of these loops of ( N 1) segments we use (23) for the configuration count normal to the surface and (27) for the configuration count parallel t o it. For all the (Ps' - 1) loops, which are attached a t PSI points, we thus have ;(jvpsf)= 2 3 m ' - w x
introduced in the clt'ort to eliminate unwanted configurations. Only with the last term in each case is the final answer for fil established. If we now make use of the fact that
we can express the above series as expansions in powers of y .
-
f?o =
n, 2.8371
= Ql[lfi2 =
-
-
Yol. 86
6,=
+ 5.2y2] O a [ l - 4 . 0 8 ~+ 8 . 5 ~ 8y3], ~ etc. %[1 - 3.677
-
It follows from (do), however, that for large N the
