M aterials Chemistry and Phy sics, 15 (1987) 51 511 zyxwvutsrqpo l-524 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON A GRAPH REPRESENTATION FOR CROSSLINKED POLYMERS AND THE TOPOLOGICAL GELATION CRITERION zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML W. KLONOWSKI and A.E. HAMIELEC, McMaster Institute for Polymer Production Technology, McMaster University, Hamilton, Ont., L8S 4L7 (Canada) Received September 15, 1986; accepted October 15, 1986 ABSTRACT A representation of a crosslinked polymer material by a graph is proposed. Considering connectivity of this graph one may easily introduce notions necessary to formulate the Topological Gelation Criterion for the system with which the graph is isomorphic. Then the condition for the critical sol-gel transition point Is formulated in terms of this graph. The number of molecular aggregates in the system may also be easily calculated from the cyclomatic number of the graph. INTROOUCTION In a previous paper [l] we have discussed the usefulness of graphs in analysing structure-propertyrelationships of amorphous materials. In the present paper, we give an example of the isomorphism between a crosslinked polymer system and a graph. It enables one to define and to calculate a probabilistic representation of the system (cf.[l]) which is subsequently used to formulate the Topological Gelation Criterion (TGC) to calculate the critical sol-gel transition point. In 121 we have calculated a probabilistic representation 8 = k$(hi)) (I) (cf. eqn. (4) of [l]) for a crosslinked polymer system with two different types of active groups. In this case, the probabilistic parameters qkm are the contributions of different categories and classes of crosslink interaCtiOnS, 0254- 0584/87/$3.50 "kc and nkt, and the contributions of non-bound macromolecules, 0 Elsevier Sequoia/Printed in The Netherlands 512 zA and zg. According to the main assumption of the Probabilistic Theory of Crosslinked Systems (PTCS), the properties of the system depend on the actual structure, mainly on the defects in network connectivity [1,31. The topology of the system is characterized by two parameters, hi - the crosslinking density index L!,and the fraction of crosslinks closing cycles p (2,31. So the probabilistic representation in the case under consideration is given by where k = 0, . . . , 4; m = c,t; I =A,B. I3itself may be expressed as a function of .Qusing the Principle of Macroscopic Uniformity 12.31. The total number of crosslinks, and so the parameters e and p, may, however. change with time and then the state of aggregation (the contributions of different categories and classes of crosslinks) changes too. The corresponding changes of the properties of the system may be quite dramatic if the critical gel point is crossed. A very interesting case would arise if the crosslinking process has an autooscillatorycharacter, or if physical properties have a feedback effect on the crosslinking process, which may then induce an oscillatory behaviour eventually leading to a new type of dissipative structure - the temporal sol-gel structure i1.31. Another interesting possibility arises when the distribution of reactive groups between primary macromolecules changes simultaneouslywith crosslinking, e.g. by polymerization-degradationprocesses. This also influences the critical gel point and may serve as a regulatory mechanism for sol-gel transition. As the PTCS is not based on assumptions about the equilibrium state of the system, such cases may be treated by using this theory. We shall start our considerationswith the same system as in 121 - two kinds of macromolecules,one bearing reactive groups of type A and the other groups of type B. From a topological point of view, however, the situation is exactly the same when the crosslinking species are small molecules. Therefore the PTCS can also predict the critical gel point in systems in which multifunctional polymerization takes place. The critical points predicted for such systems by the Topological Gelatlon Criterion, derived from PTCS, are in better agreement with experimental data than predictions given by any other existing theory [4]. 513 MODELLING OF POLYMER SYSTEMS BY COLORED GRAPHS There exist numerous treatments on graph-theory and its applications. Unfortunately, the terminology used is often very different. We have decided to use here one systemized by Essam and Fischer [51 which we will briefly describe, together with some necessary extensions and generalizations. An abstract qraph Is a set-theoreticmathematical object, having certain topological properties, and which ought to be distinguished from Its possible geometrical representations,e.g. by some figures. A crosslinked system may be thought to be a three-dimensional representation of abstract graphs which are isomorphic to it. And oppositely, a graph serves as a representation or a model of the system. Our definitions ought to be distinguished, however, from the so-called graph-like state of matter introduced by Gordon and co-workers (e.g. Ref. 161). So the system under consideration may be treated as a representation of an abstract undirected and disconnected graph without loops 151, which will further be called simply the graph. Crosslinks are isomorphic with graph vertices and polymer chains are isomorphic with graph edges. However, to take into account uncrosslinked primary macromolecules and pendant chains, one must suppose that the ends of all primary macromolecules are also isomorphic with graph vertices. The vertices isomorphic with free chain ends of different kinds of macromolecules are distinguished from those isomorphic with network junctions, by giving them different -9 colors say A, B and J, respectively (vertex colored graph). Different colors, say A and 8, are assigned also to edges that are isomorphic with chains belonging to macromolecules of different kinds (edge colored graph). Generalizing the definition of a vertex valence 151, we will say that a vertex has a _J-valence equal to WJ. if it is connected to other vertices of color J (j-vertices) by WJ of its incident edges. Similarly, one defines A-valence and B- valence. Of course WJ + WA + WB = W (3) We shall denote crosslinks in the polymer system by capital letters while corresponding vertices in the graph by small letters. A tilde I-' over a symbol will denote the number of corresponding crosslinks or vertices. So, a crosslink Jkc or Jkt is isomorphic with a j-vertex having WJ = (4-k) and the total valence w =4. For a- and b-vertices, w is always one (pendant vertices or l-vertices of color A and B, respectively) and either WJ = 1 or wJ = 0. In Fig. 1 we show the classes and categories of graph vertices isomorphic with crosslinks under consideration. 514 The vertices having ~~23 are called j-nodes (or just nodes or principal points), the vertices with WJ = 1 or WJ = 2 are called antinodes. A cut-vertex (or articulation point) is a vertex, deletion of which produces a graph which has one component more (by breaking a connected component into two disjoint ones). Multiedges, namely -doublets i.e. two parallel edges linking two j vertices, are possible, whereas loops are excluded in the graph under consideration. A m, i.e. a single edge incident with only one vertex, ought to be distinguished from a c~&, i.e. from a sequence of edges such that initial and terminal vertices coincide. The maximum number of independent cycles is called cvclomatic number, c. c=e-p+n (4) where e denotes the number of edges, p the total number of vertices and n the number of components the graph is composed of (for a connected graph n = 1) 151. One ought to note that a dimer in the graph-theory I51 is the graph isomeric to a sinsle non-bound macromolecule in a polymer system. If a possibility of misunderstandingexists, a (macro)molecularaggregate consisting of two primary (macro)moleculeswill be called a molecular dimer. Also, the graph-theoreticalterm &i~ ought to be distinguished from a molecular chain, i.e. a fragment of a macromolecule between two crosslinks or a free end of a primary macromolecule. We define suooression of a vertex of valence w = 4 (cf. Fig. 2) as splitting the vertex in such a way that two e-vertices, called g-vertices gA-vertex (incident with two edges of color A) and gB-vertex (incident with two edges of color B), are produced. Then one suppresses the two g-vertices according to the Essam-Fischerdefinition 151. The g-vertices into which a j-vertex is split are isomorphicwith reacted qrouos in the corresponding polymer system 1731. Under each category and class of j-vertices shown in Fig. 1 we give its "composition"of g-vertices (cf. Table I and 121). Calculation of probabilistic representation,Sc (eqn. (l), 121) is directly based on this composition. Free reactive qrouos in the system are represented by e-vertices called r-vertices (rA-and rB-vertices, respectively). A polvedqe is a set of edges of one color linked together by r-vertices so that after suppression of all j and r vertices it becomes a dimer graph, consisting of a single edge with two l-vertices on its ends. A polyedge is isomorphic with a primary macromolecule (polyelement) 191. There are NA and NB polyedges of color A and B, respectively, in the graph under consideration. 515 The definition of the representation of a structure by graph zyxwvutsrq [Zl ought to be modified for representing crosslinked polymer systems to be in confo~ity with our general theory of Systems with Uiscrete Interactions (SDI) 191. According to SD1 theory a cross-interactionis defined as the set of linked elements; in the case under consideration a crosslink is a set of two linked (reacted) groups, one of type A and one of type B. Analogically, an internal interaction is defined as the set of either linked or free elements belonging to one polyelement; in the case under consideration a molecular chain connects two A groups or two B groups, reacted or free, belonging to one primary macromolecule. So crosslinked polymer structure is made isomorphic with a graph G(V,E) by the relation internal interactions & cross interactions e edges, E vertices , V (Ea) (5b) and G itself will be called an inverse cross representation (in-representation)of the system. The probabilistic representation, SC (eqn. (2)) remains, however, unchanged. The full isomorphfsm between the crosslinked polymer system, SD1 and a model graph is given in Table I. GRAPH-THEORETICALAPPROACH TO NETWORK CONNECTIVITY Let us recall some definitions introduced by one of the authors (7-91 to formulated Topological Gelation Criterion. These may be easily understood only while using graph theory. We find it convenient to introduce the notion of Gellv-CaDable Crosslinks (m) (formerly called 'gelly-effectivejunctions' [7 zyxwvutsrqponmlkjihgfedcb 1 , which, however, has led to some misunderstandings). A crosslink is a GCC if it links two fragments of the system, consisting each of more than one primary macromolecule, such that two fragments become separated if the crosslink under consideration is broken. The fraction of GCC can be calculated in a straightforwardmanner using the PTCS. GCC are a subcateoory of the category of crosslinks forming ~lecular trees [l-31. For example, in the case under consideration, the crosslinks in molecular dimers (Jet), crosslinks linking 'tassels' (pending chains) (J3t and J2,2t) as well as crosslinks closing Cycles zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ (Jk', k - 0,1,2) are not GCC. 516 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA X Bt At xx xx 3 I gozyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA go I ’ .t zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ,t J, JO .t .t Jz., J 2.2 \/i/ x . + -. .t--\ \ + / x Fig. 1. j-vertices categorization and classification. j m denotes a j-vertex of category m where m=t (tree forming), m-c (cycle closing) and of class k (k=O,l, . . . where w denotes the total valence of a is further subdivlded into twotclasses (cf. Fig. 2 A- and rU- vertices are not shown. Under the given category and (sub)class of j-vertices its m- vertices is given. Explanation of symbols is given in zyxwvut X .t X J 2.2 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML jil zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA a. / 4 -\ -LT I + zyxwvutsrqponmlk ii b. /-\ P 4 I + %I bi zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED C. Fig. 2. Examples of a j-vertex splitting and suppression. (a) Different categories (t,c) and (sub)classes of vertices classified into the 2nd class (k = 2), i.e. having WJ = 2. A dashed line denotes a close cycle (cf. Fig. 1). (b) By splittirtgany jk-vertex two e-vertices are produfemd:one grAm-vertex, vertex; note the difference between g -vertices, and one g k reacted groups,and rI-vertlces,isomorh'Fcwith free reactive isomorphii w?ih groups. The fragments which have been jointed by a jP -vertex become disconnected whereas those joint by a jc-vertex are still connected throuoh a chain of other edges and vertices.. _ ICI In the second steb of suooression of a &vertex everv a,Im-vertex II = A,B) produced of the just splitted j-vertex is in turn supp%ssed (this-is a normal procedure of suppressing any 2 vertex) [51. It is easy to see that suppression of any jc-vertex in G(V,E) dimlnishes the cyclomatic number by one without changing the number of connected components, whereas suppression of a jt-vertex does not change the cyclomatic number but increases the number of connected components by one. l-vertex B (b-vertex) l-vertex A (a-vertex) L grBm-vertex grAm-vertex B 0 any vertex j-vertex LO o--O B-edge or B-polyedge A-edge or A-polyedge e--o a chain of cycle-closing any edge or any polyedge .#-a, edges. + 518 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Only GCC may constitute parts of molecular tracts going to 'infinity' (I.e. to the sample boundaries). The existence of such tracts is an indispendableproperty of critically and supracriticallybranched systems (e.g. of gels). Any reacted group involved in a GCC, is called Gellv-Capable Group(s). (a), A primary macromolecule is called Gellv-CaDable Macromolecule if it has at least two reacted groups involved in crosslinks. For example, in the case under consideration, the pending chains, the macro~lecules forming dimers and, of course, non-bound macro~lecules are not GCH. The notion of Gelly-Capable Crosslinks may be easily explained only on the graph-theoreticalbasis in the following way. In any connected component of the graph representing the system under consideration (Fig. 3a), one suppresses the maximum possible number of j-vertices, i.e. such that the component is not decomposed into smaller subcomponentsbut suppression of any remaining j-vertices causes a decomposition, In such a way each component is transformed into a &gg (Fig. 3b) and the cyclomatic number of the transformed graph becomes equal to zero. All vertices which may be suppressed in such a way are cycle-closins vertices and they are isomorphic with the crosslinks of category c [2,3,71. The suppresslon of one cycle-closing vertex diminished the total number of vertices, p. by one, the total number of edges, e, by two, while the number of components, n, remains unchanged; so the cyclomatfc number (cf. eqn. (2)) is diminished by one, i.e. each crosslfnk of category c closes one independent cycle. The vertices which still remain in the transformed graph, tree-forminq vertices, are isomorphic with the crosslinks of category t [2,31. Now one suppresses all t vertices having J-valence, WJ, equal to 0 or 1, as well as those having either WJ = 2 and WA = 2 or WJ = 2 ani wg = 2. The SUppreSSiOn of a tree-forming vertex never changes the cyclomatic number of the graph since it diminishes the number of vertices by one and the number of edges by two, but simultaneouslyincreases the number of components by one (b. eqn.(4))+ The vertices which still remain, Q. wg =1 (a. those with WJ = 2, but WA = 1 and of subclass j2,1t) as well as with WJ = 3, WJ = 4 (of classes jlt and jot) are GraDh Exuandinq j-vertices, GEJ, isomorphic with gelly-capable crosslinks, GCC (Fig. 3~). It means that the vertex representing GCC is a cut-vertex befng either a j-node or an antinode with wA<2 and wg<2 (Ffg. 1). The A- and E-polyedges which still remain connected after suppression of all vertices not being GEJ. are called Graph-Exoandina Polvedaes, GEP, and are isomorphicwith primary macromolecules being GCM. 519 Since each j-vertex may be considered as a set of two g-vertices and so as isomorphic with two elements {reacted groups) joint together 19, 101, the g-vertices involved in GEJ are called Graph ExDandina a-Vertices, GEG, and are isomorphic with Gelly-Capable Groups, GCG (Fig. 3d) (cf. Table I). Table I. Isomorphism between (macro)molecularsystem, General System with Discrete Interactions and graph. Also includes the list of short-hand notation. Crosslinked Polymer Systems (and General System with Discrete Interactions) Graph crosslink (cross-interaction) j-vertex reactive group of type A or B (free element) rA-vertex, rB-vertex (2-vertex) reacted group of type A or B (linked element) grAm-vertex, grBm-vertex (2-vertex) free end of primary (macro)~lecule of kind A or B a-vertex, b-vertex (l-vertex or pendent vertex) macromolecular chain l.e. fragment of a primary macromolecule between two crosslinks or between two reactive groups or between a crosslink and a reactive group (internal interation) edge free chain end pending edge primary macromolecule (polyelement) polyedge functionality of a crosslink, K total valence of a vertex, w number of chains going from the given crosslink to other crosslinks J-valence of a vertex, wj GCC, Gelly-Capable Crosslink GEJ, Graph-Expanding GCG, Gelly-Capable Group GEG, Graph-Expanding g-Vertex j-Vertex GCM, Gelly-Capable Macromolecules GEP, Graph-Expanding Polyedge Note: The notions used for General System with Discrete Interactions [9l are given in the first column in parentheses. zyxwvutsr 520 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA GC zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR b. G9 GP C. d. Illustrationof the concept of Graph-Expandingj- and g-Vertices Fig. 3 and Polyedges. (a) Graph G isomorphic with the macromolecular system under consideration. Symbols are the same as in Figs. 1 and 2. For simplkity rA- and rB-vertices are not shown. (b) By splitting in G the maximum possible number of j-vertices without changing the number of connecting components (e.g. the vertices shown by arrows) one obtains graph Ge having only tree-forming vertices while having no cycles. ("1 BY SUPPIessing in Ge all j-vertices which having been classified as jJt, j3 and jE 2 ,and subsequent removing of all dimer components (which have been either present in G or 'produced' by the suppression procedure) one obtains Gg. The j-vertices and polyedges still present in Gg are GEJ and GEP. respectively. (d) By splitting any GEJ one obtains two GEG. The Gp obtained In this way consists of not-connectedpolyedges being GEP,bearing at least one GEG each. 521 TOPOLOGICAL GELATION CRITERION As the number fitof crosslinks of the category Jt is just a minimal number of crosslinks necessary to obtain a given state of aggregation and each crosslink of category Jc closes a cycle, the cyclomatic number may be identified with the number Jc. The number of vertices, p, is identical with total number of crosslinks plus twice the number of all primary macromolecules (i.e. the total number of a- and b-vertices). From eqn. (4) and from the definitions of 4 and p [2,3] one obtains the number of connected components in the graph G(V,E) and so the number of molecular aggregates in the crossllnked system under consideration n = [(NA + NB)/ZI 12 - e (1 - S)l (6) This number includes eventually the 'infinite' component of the graph which is isomorphic with the gel in the polymer system. The condition for such an infinite component to exist, called Topoloqical Gelation Criterion, was first stated by one of the authors [7,81. It is based on the concept of topological connectivity, formulated in terms of GCC and GCM. One defines the Reduced Crosslinkino Density Index (L), as the number of GCG per GCM (cf. 131). L = $/Ng (7) where zg denotes the total number of GCG and Ng the total number of GCM. The L is, of course, a function of the crosslinking density index, @, and through it a function of time. It reaches a critical value Lcr at the gel point, i.e. for crosslinking density index @ = f!cr 18,101. The TODOlOoiCal Gelation Criterion (TGC) 13, 71 says Lcr = L(f?cr)= 2 (8) In the system under consideration, only reacted groups belonging to crosslinks Jot, Jlt and J2,1t are GCG. To calculate the number of GCM it is necessary to subtract from (NAzA + NBzB) macromolecules bearing at least one crosslinked element those involved in molecular dimers (in each dimer there is one macromolecule A and one B) and those forming pendant chains. One obtains RCL in the form 522 L= -t t t zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG 2J@, + nt + n2,1) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA(9) zyxwvutsrqp UNAzA+ NBs)-$2n:+ni +&I where $and cdenote the total number of crosslinks and of reacted groups in the system. IntroducingRCL in the form eqn. (9) into TGC (eqn. (8)) and taking into account the definition of parameters p and normalization conditions (cf. Ref. [31), one obtains TGC in the form (2nk + n”,-ni - 2ni-ni) (11) = Cl t=tcr For a tree-like crosslinked system as considered by other authors (cf. 161) nkC are by definition equal to zero and so those theories have necessarily limited applications. Equation (11) is a transcendental equation for critical crosslinking density index, @cr. CONCLUDING REMARKS We have used Topological Gelation Criterion previously without introducing graph representationfor the crosslinked system under consideration. However, the concepts of Gelly-Capable Crosslinks, Groups and Macromolecules,which are absolutely essential in definition of the Reduced Crosslinking Density Index, and so in formulation of TGC, may be easily explained only when graph representation and the notions of Graph-Expanding j-Vertices, g-vertices and Polyedges are introduced. These concepts may be even more easily formulated and understood if the system is modelled rather by a hypergraph [lO,lll than by a graph. It, however, needs introduction of the notion of a hypergraph, which is much less known than that of a graph. We have demonstrated that while using TGC one may predict the critical gel point in different systems with better accuracy than using other existing theories [41. We have used a graph with two-color vertices and two-color edges, which is isomorphic with a system with two different kinds of primary macromolecules,A and 8. For the system with only one kind of primary macromolecule, it is more complicated to give graph-theoreticalexplanation 523 to GCC and GCM. zyxwvutsrq But as connectivity of the system is a topological property and does not depend on the colors of vertices and edges, one may imagine that the system with one type of groups is a "limit" of systems with two types when the types become more and more similar to each other. A paradox like the Gibbs Paradox in gas theory does not arise as we are not interested in counting distinguishable permutations of vertices and edges but only in total numbers of GCC and GCM in any representation of the graph. Another way of dealing with a system with different types of groups consists of giving to any polyedge a different label (e.g. a number). Then all edges which are isomorphic with molecular chains formed by fragments of any crosslinked macromolecule may be distinguished from those isomorphic with molecular chains formed by any other macro-molecule. The introduction of labelled polyedges allows much more detailed discussion of the structure and properties of the graph isomorphic to a crosslinked system. However, in the case under consideration,which is a special case with only two different labels (namely A and B), we do not need such detailed discussion. In papers to follow, we shall consider application of the general model introduced in this paper to well defined polymer systems, both equilibrium and dynamical 1121. Because of a topological nature of the presented theory it may be applied to systems not being in thermodynamic equilibrium while existing theories could be applied practically only to equilibrium systems 1131. ACKNOWLEDGEMENTS Thanks are due to Professor W. Brostow (Drexel University, Philadelphia, PA), the late Professor P.J. Flory (Stanford University, Stanford, CA) and to Professor J.F. Rabek (Royal Institute of Technology, Stockholm, Sweden) for discussion and comments concerning this paper. REFERENCES 1 W. Klonowski, Mater. 2 W. Klonowski, Z. phys. Chemie. Leipzia. 266 (1985) 927. (1986) 581. 3 W. Klonowski, J. Appl. Phvs., 58 (1985) 2883. 4 W. Klonowski, Polvmer Bull.. 10 (1983) 567. 5 J.W. Essam and M.E. Fischer, Rev. Mod. Phvs., 42 (1970) 271. 6 M. Gordon and W.B. Temple, J. Chem. Sot.. A (1970) (5) 729. 7 W. Klonowski, Rheol. Acta. 18 (1979) 442, 667, 673. zyxwvutsrqponm 524 8 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA W. Klonowski, Bull. Cl. Sci., Acad. ROY. 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