,JiiIy, I955
POLYVICR C(H.\IS CC)NFIGITR.~TION r\TE.\R
.i h U N D . - \ R T r;',XlCRTING
633
Fonces
Two analyses of the mother liquor a t 21.0 and the gap disappears from the isotherms a t iibout 25". There are no other properties of the q-stpins \vort,liy 93.0 may tie gi\,eii liere to il1ustr:ite t,he composition of special mention. ('rd)le 11).
R p c e i r w r i Fubr,uat,u 5 , ID:i.5
The configuration of a flesil)le polymer chain, liuilt up by the successive deposit ion of single segineiits, ne:w:L ptiwp h o ~ ~ t ~ d nry exerting forces oii the individual segments is studied. It is shown that the end-to-end distauce pi,olx~l t i l i t ) . tlistji4iutioii satisfies instead of a simple diffusion equation a Fokker-Planck equnt,ion whose convective terms we propoi,t ioit:LI t o the forces exerted by the boundary. When these forces are small, solutiolis by pertui81,ntion series ai'e given, w h i k l o r Inrge forces an approximate solution is derived in the case of attraction. The effect of the chain configurntion on such suIIace properties as the adsorption of polymer from solution, the stabi1iz:itioii of colloidal suspensions by dissolved macromolecules, the sui*facetension of polymer solutions, etc., is treated.
I. Introduction Theoretical studies of the adsorption of flexible macromolecules onto solids from their the surface tension of polymer solutions3 as well as certain other surface properties of polymer solut i o n ~show ~ that these may depend strongly on the configuration of the polymer chain near a phase boundary. I n the case of adsorption, preliminary experimental investigations show that such is indeed the case and indicate rough qualitative agreement between theory and experiment,.b I n the absence of strong forces exerted by the phase boundary, the configuration of the polymer chain can be assumed to be given by an appropri:ite solution of the diffusion equation with a boundary condition corresponding t o perfect reflection and neglecting disturbing effects arisiiig from excluded volume. I n systems .without pronounced polar character or polymer-substrate specificity, the forces act,ing on the polymer are of the order of kT, that is, sufficiently weak t o produce a t most a small pert)urbiition in the random coil configuration of the polymer. We shall estimate the magnitude of this perturbation. On the other hand in certain systems strong forces will be acting on the segments of the polymer chain leading to the failui*e of a simple perturbation t,reatment. Such a n instance could well be expected t o occur in biologically important polymer substrate systems, e.g., antigen-antibody reactions, in which mutual specificity plays an important role. We shall present a treatment which applies in first approximation in such cases (strong forces) for sufficiently large polymer chains. The precise specification of a polymer chain of t segments in ( 1 ) H. L. Friscli, R. Sirnhaand F. R. Eirich, J . Chem. Phys., 21, 365 (1953); R. Sinilia. H. L. Friscli and F. R. Eiricli, THISJOURNAL, 57, 584 (19.53): H. 1,. Frisch and R. Sirnha, ibid., 58, 507 (1951). (2) R. Ullinan, paiier presented at t h e 124th rneoting of t h e Ainerican Clieinical Society, September 0-11, 1953, Chicago, Illinois, (3) Fuiiiio Onsawa a n d Slio Asakura, J . Cheni. Phys., 22, 1255 (1954). (4) Such as for instance the stnhilizntion of colloidal swpensions. C/. E. L. hfackor a n d J. H. van der Wnals, J . Colloid S c i . , 7, 585 (1952) as well a $ reference 3. (.5) W. Heller a n d T. L. Pugli, J . Chrtn. PILUS.,2 2 , 1778 (1954), as well as papers quoted in referenres 1 .
such a forct: field would require the introduction of the joint probability distributiori of all t segment's, specifying the location of each segment. I n most' applications such detail is unnecessary, interest being centered on the probal)ility distribution of endto-end distances w(z, y, x; t ) where 5 , y, z are t,lie coordinates in a coordinate syst,ern of the tth segmeut a t whose origin the first segment of the polymer chain is located. We shall try to estimat'e this probability assuming the following. (1) That w(x, y, x; t ) is not affected by the force field in the x and y directtion but merely along the x directmion(being the vertical distance of t'he segment, from the boundary), so t,hat ,u(.r,I/,
2;
I) =
UJo(.z;
t ) lao(!/; I )
742;
1)
(1)
where wo(:c; t ) = ( 2 ~ f t ) - ' / eq: * ( -r2/4ft) and f is the segment diff usion coiistaiit, in the "rnndom flight" analogy. (2) The potential T'([), consist,ing in general of both attractive and repu1sii.e terms, act's on every segment s, I s t , separated a distance 4' from the phase boundary. (3) The boundary is impeiietrable to all polymer chains. (4) The depositfed polymer chain can be thought to be built up of successive depositions of the polymer segments and for simplicity but without real loss in generality the first segment is assumed to lie a t x = 0, Le., on the boundary. (5) We shall neglect certain complicating effects such as those arising from excluded volume c,oiisiderations, etjc. Let a segment be placed in a certain configurntJion a t a dist,ance $ from the bomidary. The forw acting on this segment is obtained from the grndieiit of the potential V ( $ )as
<
hertoward or away from the phase boundary. Assumptions ( l ) , (4) and ( 5 ) guarantee that the location of the Pi segment depends only on the location of the t-1st segment and thus w(x; t ) is found as a solution
H. L. FRIRCH
(334
of a typical “random flight” integral equation whose kernel function is not isotropic. It can be shown6 that for sufficiently large t among other conditions this integral equation may in turn be replaced by the Smoluchowski equation.1,6
I‘d. .59
first few momeiits of w ( x ; t ) , ~rhichare incidentJnlly linearly related ns can be seen directly from eq. 5 011 multiplication by x and integration from zero to infinity, viz.
where ~ ( z ) the , “drift velocity,” gi\*es the magnitude of the drift, of the t t h segnieiit located a t x, \rit,h = ,f[10(0; 1 ) - A < 1 / ( 2 J > ~ ] v(x) posit,i\.e if t,he forces exerted by the phase Similarly one finds for the second moment , bouiidaq. at x itre attracti\.e. Then “drift velocity” ~ ( x )satisfies the usual Langevin equation6 ( c j . eq. 2 ) Both these equations show that as expectfed stu ( s ) = P F ( 2 ) = A,fg(z) (4) tractive forces exerted by the boundary decrease where 0 the segment mobility is equal to f / k T and the extension of the chain as measured by the avereO/kT = A. Thus the larger is B~ the larger will be age and average square chain length by changing the deviation of w(z; t ) from w&; t ) which is a the functional form of the distribution and secondly solution of a simple diffusion equation obt,ained by by decreasing these by the terms t and set’tingX equal to zero in t.. For a t least small values of X the asymptotic estimate < x > t g ( < x > + ) for t should suffice. Since for small A, t is of the order of obtained by substituting eq. 4 in eq. 3. The de- t i / ?(cf. eq. 7a) say < z > ~ = at‘/?, this and eq. 7b gree of flexibility of the polymer chain also affects should allow us t o estimate the effect on the polythe magnitude of the drift term since v is propor- mer chain configuration of the range of the forces tional t o f and the latter decreases as the flexibility exerted by the boundary. Thus if these, i.e., g ( x ) , of the chain increases.’ Equation 5 has t o be fall off7as A ( x - R)-a we find solved under the initial and boundary conditions (cf. assumptions 3 and 4). w ----f 6 ( x ) as t ---f 0 (5‘) showing that the correction term becomes negligible for sufficiently large t and if 6 > 1 which will + xgw = O a t 2 = Ofor 1 > 0 generally be the case; in particular surface disax persion forces have 6 4.’ and 11. The Weak Field, High Temperature Solution w,aw/az ---f 0 as a --f + m , t > 0 For arbitrary g(z) no explicit general solution has Before we can proceed we will have to find a suitable form for V(t) the segment potential en- been found but for the limiting case of small X = ergy. This energy will itself be given as an integral eO/X:Teq. 5 can be solved in terms of a perturbation over the surface of the phase boundary of the in- series in the parameter X. This case corresponds to termolecular forces exerted on a segment by each either the case where the forces exerted by the molecule composing the boundary. These forces boundary are very weak (E,, very small) or where the may contain besides electrostatic, inductive and temperature of the polymer system is high as for dispersive force contributions, terms which may example in a melt. Thus for X < < 1 we write m arise due t o the formation of chemical or quasichemical bonds between polymer segments and w(z; t ) = x”w,,(z; t ) (8) n=O active sites on the substrate. The form of g(z) t o be used in eq. 5 will depend on the polarity, chemical where the successive wn(x; t ) are found, 011 substistructure, etc., of the molecular species involved7 tution of eq. 8 in eq. 5 zinc1 6’, to satisfy but in the absence of strong specificity will generally be represented to a sufficient approximation by the g(x) derived from a square-well potential8 V ( z )=
+
=
-EO
m
= 0
for x < 0 for 0 Q z S R for a > R
i.c. g(z) =
a(. -
x;,z > 0
(6)
In most applications of this theory1-4 t o surface sensitive properties of polymer solutions one requires only the value of w(z; t ) a t x = 0 and the (GI 5. Cliandrasekliar, Reu. Modern Pkiis., 16, l ( 1 9 4 3 ) . ( 7 ) For a yecent review on detnils of these siirface forces see J. H. Honig, A n n . N . 1’. Acad. Sci., 6 8 , 741 (19.54). (8) J. 0. Hirschfelder, C. F. Curtiss and R. B. Bird, “Rlolecular Theory of Gases and Liquids,” J o h n Wiley and Sons, Tnc., New York, N. Y., 1954, p . 158 ff.
and
by the phase boundary a balance betweeii the diffusioii "force" with both the harmonic restoring force and the forces eserted by the boundary producing the drift velocity v(z) = f X g ( x ) must be reached as long as v(z) > 0. If these forces are very strong, X >> 1, we can neglect the cffect of the harmonic: restoring force in this bdailce. Ailteniatively phrased \ r e can coiisider this instance as one in which we are justified i n neglecting "r2~1iclom flight" correlations in placing successive segments of the chain i n comparison with the effect due t o the force field of the boundary, i.c., dw/bt i n eq. 5 vanishes aiid we thus obtain
The successive solutions of eq. 0 for w,, can he obtained by i1iverting the Laplace tratisforms of w&; t ) and wIL(z;t ) @(,(z; p ) =
ha
e-pt
wo(z; t ) d t
The Greens' function y, - l(z, zo; p ) is found t80he e-qizo
- q,,- I ( ? ,
20;
p) = 1
+
2 [ ~
- {E-'('"
-I)
+
Q-I
2)
(0;
P)I
H ( r o - :j
An appropriat,e solutioii of this equation is
+ + p-qtr
-
:111
H ( ? - zo))
since it satisfies the boundary conditioiis ;it zero, approximately t,he one a t infinity and c a i l t)e normalized
29
\vith H(.c) t'he Hea\.isitle unit function. Eyuations 8 aiid 10 specify the desired solution for small A, for example if g ( x ) sa.tisfied eq. (j one finds using the nbo\-e solution that 2 [1 ( 27rf / j'/? = ~ " ( 0/ ;) [ I -k Xe
w(0; I ) =
~
+ -R2"t]
+ O(X2)
so that,
(I Ij
+ O(h*)
If R is small, R = 2 ; t,hen the above result shows that the magnit,ude of the correction depends pri- Substituting eq. 2 int,o eq. 131, we find marily on eo the depth of the potential well (cf. ey. 6). Although €0 will vary considerably tlepeiidiiig on the nature of the substrate-polymer syst'ein under consideration a rough idea of the order of magnitude of this entity can be obtained from the Corn(1Bc) parable eo values found for a number of pure vapors from the second virial coefficient.' Thus in the absence of specificity and pronounced polar character I n deriving eq. 13c we ha1.e assumed that the strong boundary forces remain strotig over the whole €0 in A could be approximated by t,he values of simple hydrocarbons, e.g., to/X: N 244'1L for ethaiie, ratige of distances accessible t o the tth polymer segeta. In the solutioii this value should be furt'her ment of our chaiii. Should this not he the case as reduced i n i%\v of the shieldiiig by t8he solreiit for example for a very deep hut, narrow square well molecules. The perturbation procedure is t,hus potential eq. 13c should he replaced in first npproxijustified and the magiiit'ude of the correction which mation by the more general relation has to be introtIuced7nto w(0; t j \ T7(rj - T'(Oj/. esp . - _AT __ H ( K - 2) 7uo(z; f ) H ( i - R ) and t8hemomeiits \vi11 not, exceed a numerical factor of two at w(i;I ) = (14) \ I,*(?) - V(O)( 1 hm{eXl) ;-~x,jp,H(/? - Z ) +?/!a(?; 1 ) H ( 2 - R j i (12 room temperature or above. As polar itiflueiices increase the treatment presented abo\,e becomes less satisfactory where w&; 1) is the unperturbed prohahilit,y tiistrisince E ~ / / Cfor methyl chloride is about 469"Ii. \rhile bution which holds outside z = R t,he range of tjhe for water vapor in which the possibility of hydrogen forces eserted on t'lie tth segmeiit, by the phase bonding exists c o / k is 1260°1iontreatment must fail. IV. Applications and Conclusion 111. The Strong Attractive Field Solution The applications of t,he foregoing theory are dit o t,liose For lalge .\.slues of eo//; a11alterllate approsimat,e verse and we shall restrict 0111' tlisc.r~sr;;ion solutio11of e ( ] .5 ]vi11 be preselltetl !\.hiah llolds only mentioned i n the introduction, hi pai*ticulaiatJhe if the forces esert,ed t ) l T 1)outidary Are T-ery st,rOllglJr aclsorpt'ioii of flexible macromolecules from t'heir attract,i\re. 111 t,lle ahsetlce of esterll:tl forces olie solut'ion. -4s has beeti pre\-iously nho\rii' tflieprohcall represelit the 1ltlperturhed coiifigur~tiotiof t,Ile ability that t'he T~~ segment' of the polymer chain polymer chaitl as resiiltiiig from a statioiiary I)aIaiice adheres to the phase houiidary, p ( r ) is gir.eli by (bwlbt = 0) between the effec+t,s of diffusiotl of the wo(.r, y, 0 ; r ) p d p ~ I $ (15n) P ( T ) = 4 1 - 0) chain ends arid a harmonic restoring force (replacing the bw/bt term) with a force cotistatit propor= a0(l wo(.x; T ) U ~ ~ ( T~) Jw ;( O ; T ) p r t p d + tional t o LT and inversely proportion:tl t o f . This suggests that even in the presence of forces exerted = 2 4 1 - e ) w(0; r ) [ l - e - ~ / U l ; 1 S T 1
,
,
+
so""
OjKJozr
where 13 is the fraction of the surface of the phase boundary covered, a. is the probability that if a segment touches it adheres, p2 = .c2 .!J~and 4 = tan-l (y/x). It is by way of this relation that the configuration of the chain influences the adsorption of the chain through the indicated proportionality t o w(0, t). For weak boundary forces w(0, t ) is independent of the range of these forces (6 = 4) as long as these decrease a t least as the inverse square of the distance between the segment and the boundary (cf. eq. 7) and for non-polar neutral polymer segments and molecules of the substrate w(0, t ) is given by eq. 11 in first approximation. This leads' toanaverage t = p = [2a0(1- $ ) / ( ~ f t ) ~ / 2 ] [l A ] . The average number of anchors < v > = pt = < v > [I ~ A ] is still O(t''2) and the adsorptioii isotherm is unchanged in form from that given in reference 1
+
+
+
~e 1 - e
(KC)' = (RC)1/o[lf
XI
(151))
most by a factor of two (A > kT it follows from eq. 14 that for moderate R, w(0; t ) = l / R and hence < v > = pt = constant (1 - $)aot/Rby virtue of eq. 1Sa. This result means that sufficiently strong attractive forces of moderate range cause the chains t o deposit completely onto the substrate surface. In such a case the amount of polymer removed from even very dilute solution almost equals the saturation value. A4sthe range of the boundary forces decreases the result given above must be modified somewhat since w(0; t ) is no longer completely independent of t , nonetheless, to a first approximation < v > is still proportional to t rather than t 1 / 2 . The implications of the foregoing both for the mechanism of reinforcement and stabilization of colloid particles follows directly from previous discussions of the increase of the parameter ,lat4,5
neglecting activity correctioiis where C is the concentration of polymer in solution and K is the known affinity constant. The increase in < v > is a t
The author is indebted t o Professor Robert Simha for having suggested this problem and many clarifying discussions.
THE SORPTION OF WATER VAPOR ON JIAGNESIURS OXIDE BY R.I. RAZOUK AND R. SH. MIKHAIL Chemistry Department, Facully of Science, Eiia-Shams Univeysity, Abhassia, Cairo, Egypt Receiued Februarg 6, 196.6
Sorption-desorption isotherms of wat,er vapor were determined a t 35" on seven specimens of magnesium oxide prepared by the dehydration of brucite a t 350, 500, 650, 800, 950, 1020 and 1100" with the aid of an electric sorption balance. The sorption isotherms are sigmoid, and definite limiting sorption values are invariably attained a t saturation vapor pressure. The oxides prepared at or below 650" are rehydrated stoichiometrically a t saturation vapor pi'essure, but as the temperature of dehydration is raised, the uptake of water a t saturation diminishes, and, ultimately, when brucite is dehydrated a t 1100", the oxide obtained is "deadburnt" and does not take up the least amount of water froni the vapor phase. In all cases, appreciablc hysteresis occurs, aud thorough evacuation a t room temperature of the rehydrated oxide removes only almost two-thirds of the water content a t saturation, while the remaining one-third can only be expelled by outgassing a t higher temperatures. Resorption isotherms are also sigmoid but distinct from the initial sorption isot,hermsthough the two curves join together a t the saturation point, forming thus a perfectly reproducible desorption-resorption loop. It is suggested that the water uptake is not merely a surface phenomenon, brit rather a bulk effect, and that the water molecules are held up in the oxide lattice with binding forces which have two different strengths each of which being characteristic of the orientations in one of the two lattices of magnesium oxide crystallites described by Garrido.
Introduction The sorption of water vapor on oxides and hydrous oxides has been studied estensively in certain cases especially on oxide gels like silica, alumina and ferric oxide gels. Other metal oxide-water vapor systems have not yet been thoroughly investigated, and the properties of such systems need further systematic study. Moreover, most of the earlier work dealt with amorphous and microcrystalline oxides, the properties of which depend t o a great extent on the exact details of preparation, the treatment they have undergone and in particular on the temperature t o which they were subjected, and on aging.' The present investigation deals with the sorption of water vapor on magnesium oxide prepared by the dehydration of native brucite under controlled conditions. This system was chosen on the following grounds : (i) brucite occurs in well-defined crystnlline form of the hexagonal C G system as lam(1)
R. J. Glenn, "The Surface Clietilistiy of Solids," London, 19651.
ellae which can be readily cleaved and may be obtained in a very pure state; (ii) extensive work has been done on the structural study of the system MgO-H,O mainly with the aid of X-ray meusurements, and, in most cases, it has proved that there exist oidy two solid phases, vix., brucite Mg(OH)z, and periclase MgO, and no othersG2 (iii) the dehydration of brucite a t various temperatures has been thoroughly s t i ~ d i e d . ~ The hydration of magnesia prepared from the carbonate or hydroside has been measured by some authors. Campbell.' obtained an impure oxide by burning magnesite betiveen GOO and 800" which could be hydrated completely in 3 days. Between 1000 and 1100", tjhe magnesia suffered a change resulting in a marked decrease in its rate of hydration, and the oside obtained by decomposition a t (2) R. Fricke, et ai.,Z. nnorg. Chim.,166, 244 (1927); G. F. Hiittig a n d W. Frankenstein, iDid., 185, 403 (1929); W. Biissem and I?. I Z n berioh, Z. p h y s i k . Chem., B17,310 (1932). (3) S. J. Gregg a n d R. I. Raaouk, J. Chem. Soc., S1,36 (194S), (4) A. J. Campbell, I d . E n g . Chem., 1, GG5 (1910).
