## Friday, August 30, 2013

### The Ascent of Sap in Trees

In Chapter 1 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I included a homework problem about moving water up trees.
Problem 34 Sap flows up a tree at a speed of about 1 mm s−1 through its vascular system (xylem), which consists of cylindrical pores of 20 μm radius. Assume the viscosity of sap is the same as the viscosity of water. What pressure difference between the bottom and top of a 100 m tall tree is needed to generate this flow? How does it compare to the hydrostatic pressure difference caused by gravity?
When you calculate the pressure needed to push water (that is, sap) up the tree through the xylem, you get (Spoiler Alert!) twenty atmospheres to overcome the viscous resistance of the pores, and ten atmospheres to overcome gravity. How does the tree generate all this pressure? That is a famous old problem known as the “ascent of sap.”

Now I admit that a 100-meter tree is, indeed, very tall; taller than even the Statue of Liberty. But it is not an unrealistic example. The majestic sequoias in California reach this height. The tallest known tree, named Hyperion, is a sequoia (coast redwood) in northern California’s Redwood National and State Parks that reaches a height of 115 m. The leaves at the top of that redwood need water to carry out photosynthesis. How do they get it?

First, let us consider some mechanisms that do not work. The tree cannot suck the water up, as if it were a gigantic drinking straw. Even if the tree could produce a true vacuum at its peak it could only create a pressure difference of one atmosphere, which corresponds to a rise of water of 10 m. Another idea is that the water rises by capillary action, like a giant wick. But the height that can be reached by climbing up a tube via surface tension is inversely proportional to the tube radius, and for xylem’s 20 micron radius tubes water will rise only a tiny fraction of the tree’s height (in Sec. 12.2 of his book Air and Water, Mark Denny estimates that water would rise in xylem by capillary action to only a height of three-fourths of a meter). Osmotic pressure won’t work either, for any realistic concentration gradient. So what is the answer?

There is still some controversy, but the generally accepted mechanism for the ascent of sap is called the cohesion-tension theory. In the leaves, capillary action through very tiny channels helps pull water upwards to replace that which evaporates from the leaf surface. In the larger pores of the xylem, the water is pulled by tension (negative pressure), somewhat like a steel cable pulling an elevator up its shaft. But can water support such a tension? It can, but there is a problem. If any air is present, the system will fail. Think of a piston filled half with water and half air. If you pull on the piston, you will just expand the air as its pressure is reduced. Now, consider the piston with only 1/4th air and 3/4th water; the air still expands when you pull. In fact, if there is even one bubble present in the water, pulling on the piston will cause it to expand. Only if the piston contains no air at all will the water be able to exert a tension force. In other words, water under negative pressure is susceptible to cavitation; the formation of bubbles. Fortunately, the structure of xylem is such that bubbles cannot grow indefinitely, but get trapped in one compartment.

For more details, see “The Cohesion-Tension Mechanism and the Acquisition of Water by Plant Roots,” by Ernst Steudle (Annual Review of Plant Physiology, Volume 52, Pages 847–875, 2001). Below I reproduce his summary of cohesion-tension theory. Note that 100 MPa is 1000 atm!
• Water has high cohesive forces. It can be subjected to from some ten to several hundred MPa before columns break. When subjected to tensions, water is in a metastable state, i.e. pressure in xylem vessels is much smaller than the equilibrium water vapor pressure at the given temperature.
•  Walls of vessels represent the weak part of the system. They may contain air or seeds of water vapor. When a critical tension is reached in the lumen of xylem vessels, pits in vessel walls allow the passage of air through them, resulting in cavitation (embolism).
• Water in vessels of higher plants forms a continuous system from evaporating surfaces in the leaves to absorbing surfaces of the roots and into the soil (soil-plant-air-continuum; SPAC). With few exceptions, water flow within the SPAC is hydraulic in nature, and the system can be described as a network of resistors arranged in series and in parallel.
• Evaporation from leaves lowers their water potential and causes water to move from the xylem to evaporating cells across leaf tissue. This reduces the pressure in the xylem, often to values well below zero (vacuum).
• Gradients in pressure (water potential) are established along transpiring plants; this causes an inflow of water from the soil into the roots and to the transpiring surfaces in the leaves.
Here is an animation that nicely summarizes this process.

I find the idea of water being hoisted up a tree by tens of atmospheres of tension to be fascinating, if a bit disconcerting. This phenomenon offers a fine example of the important role of physics in biology.