Princen model of a sheared foam. A regular hexagonal honeycomb (top left) is sheared (top right) until a topological transformation occurs (bottom left). The structure relaxes back to a regular honeycomb (bottom right), but the bubbles labeled I and III have now lost contact, while bubbles II and IV have been brought into contact.
Unit cell of the Princen hexagonal foam.
Reconnection of edges following a T1 event. Newly created film has its midpoint at the former point M B . Film inherits properties from (a periodic copy of) the former film C, so that film midpoint is a periodic copy of the former point M C .
Film B length vs imposed strain. The Gibbs parameter is .
Relative increase in the strain to T1 for three different De values, and for various . A best fit power law [proportional to ] is also shown. Throughout the liquid fraction parameter is .
Deviation from equilibrium of the surfactant concentration for film B vs imposed shear strain computed for and up to the point of T1. Data have been collapsed away from the T1 by normalizing by De. Other parameters are and . The curve labeled “via Princen” utilizes Eq. (8) but assuming the film length vs imposed strain relations derived by Princen.
Net pull from films A and C in the neighborhood of a T1 depends on the angle between films A and C, and hence on the magnitude of the vector sum of the tangents . This is influenced by the finite length of film B and by the swivel of film midpoint M A .
Comparison between film B length as it shrinks on the approach to a T1 computed via numerical solution of equations (1) and (5) with and , and the analytical formula (15) noting that for this formula . Limits of equation (15) as a power law decay—Eq. (16)—and as a Gaussian—Eq. (20)—are also shown. Film lengths at finite De invariably exceed those predicted by Princen. The cut-off length for a topological transformation is 1/128.
Relative increase in strain to T1 vs , compared to analytical predictions of Eq. (25). Throughout the liquid fraction parameter is .
The state immediately after a T1, both pre- and post-mechanical relaxation.
The length of film immediately after mechanical relaxation following a T1. The , dry foam case assumes strain to matches the Princen yield strain , and matches the net pull of films A and to the maximum pull in film . Numerical results for and both with liquid fraction parameter are also shown.
Evolution of total film length (sum of lengths of films A, , and ) during the physicochemical relaxation process following a T1 (a) for , and (b) for . The open symbols on the vertical axis indicate (at the selected values) the total length at T1 before any mechanical relaxation has taken place. In each case, . For comparison, the film length immediately prior to T1 in the original Princen model is also shown (horizontal line).
Stress (in dimensionless form) vs imposed strain for a Princen hexagonal foam, and also for a hexagonal foam driven out of physicochemical equilibrium with various values of De and .
Strain to T1 as a function of in the large Deborah number limit.
Nonlinear surface tension model given by Eq. (37) for various values. The vertical lines indicate where on the nonlinear curve the physicochemical equilibrium state c = 1 lies for either (leftmost), (middle), or (rightmost vertical line).
Tensions in films A, B and C in the case where a T1 event is just barely avoided. Film midpoints M A , M B , and M C and the vertex location are also shown. The vertical scale has been exaggerated for clarity: Strictly speaking all films should be nearly horizontal in this limiting case.
Phase diagram indicating when T1 events occur or do not occur according to the nonlinear surface tension model equation (37). The leftward and rightward pulls on the vertex in each region of the phase diagram are indicated schematically: The maximum (rightward) pull on film A is ; the minimum (leftward) pull on film B is with ; and the (leftward) pull on film C is close to (if ) but close to (if ). Leftward pull exceeding rightward pull implies a T1.
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